Population-level transcription cycles derive from stochastic timing of single-cell transcription.

Eukaryotic transcription is a dynamic process relying on a large number of proteins. By measuring the cycling expression of the pyruvate dehydrogenase kinase 4 gene in human cells, we constructed a detailed stochastic model for single-gene transcription at the molecular level using realistic kinetics for diffusion and protein complex dynamics. We observed that gene induction caused an approximate 60 min periodicity of several transcription related processes: first, the covalent histone modifications and presence of many regulatory proteins at the transcription start site; second, RNA polymerase II activity; third, chromatin loop formation; and fourth, mRNA accumulation. Our model can predict the precise timing of single-gene activity leading to transcriptional cycling on the cell population level when we take into account the sequential and irreversible multistep nature of transcriptional initiation. We propose that the cyclic nature of population gene expression is primarily based on the intrinsic periodicity of the transcription process itself.

Competitive, multi-objective, and compartmented Flux Balance Analysis for addressing tissue-specific inborn errors of metabolism.

The inborn error of metabolism phenylketonuria (PKU, OMIM 261600) is most often due to inactivation of phenylalanine hydroxylase (PAH), which converts phenylalanine (Phe) into tyrosine (Tyr). The reduced PAH activity increases blood concentration of phenylalanine and urine levels of phenylpyruvate. Flux balance analysis (FBA) of a single-compartment model of PKU predicts that maximum growth rate should be reduced unless Tyr is supplemented. However, the PKU phenotype is lack of development of brain function specifically, and Phe reduction rather than Tyr supplementation cures the disease. Phe and Tyr cross the blood-brain barrier (BBB) through the aromatic amino acid transporter implying that the two transport reactions interact. However, FBA does not accommodate such competitive interactions. We here report on an extension to FBA that enables it to deal with such interactions. We built a three-compartment model, made the common transport across the BBB explicit, and included dopamine and serotonin synthesis as parts of the brain function to be delivered by FBA. With these ramifications, FBA of the genome-scale metabolic model extended to three compartments does explain that (i) the disease is brain specific, (ii) phenylpyruvate in urine is a biomarker, (iii) excess of blood-phenylalanine rather than shortage of blood-tyrosine causes brain pathology, and (iv) Phe deprivation is the better therapy. The new approach also suggests (v) explanations for differences in pathology between individuals with the same PAH inactivation, and (vi) interference of disease and therapy with the functioning of other neurotransmitters.

Gear Shifting in Biological Energy Transduction.

Confronted with thermodynamically adverse output processes, free-energy transducers may shift to lower gears, thereby reducing output per unit input. This option is well known for inanimate machines such as automobiles, but unappreciated in biology. The present study extends existing non-equilibrium thermodynamic principles to underpin biological gear shifting and identify possible mechanisms. It shows that gear shifting differs from altering the degree of coupling and that living systems may use it to optimize their performance: microbial growth is ultimately powered by the Gibbs energy of catabolism, which is partially transformed into Gibbs energy ('output force') in the ATP that is produced. If this output force is high, the cell may turn to a catabolic pathway with a lower ATP stoichiometry. Notwithstanding the reduced stoichiometry, the ATP synthesis flux may then actually increase as compared to that in a system without gear shift, in which growth might come to a halt. A 'variomatic' gear switching strategy should be optimal, explaining why organisms avail themselves of multiple catabolic pathways, as these enable them to shift gears when the growing gets tough.

Recommendations for performing measurements of apparent equilibrium constants of enzyme-catalyzed reactions and for reporting the results of these measurements.

The measurement of values of apparent equilibrium constants K' for enzyme-catalyzed reactions involve a substantial number of critical details, neglect of which could lead to systematic errors. Here, interferences, impurities in the substances used, and failure to achieve equilibrium are matters of substantial consequence. Careful reporting of results is of great importance if the results are to have archival value. Thus, attention must be paid to the identification of the substances, specification of the reaction(s), the conditions of reaction, the definition of the equilibrium constant(s) and standard states, the use of standard nomenclature, symbols, and units, and uncertainties. This document contains a general discussion of various aspects of these equilibrium measurements as well as STRENDA (Standards for Reporting Enzymology Data) recommendations regarding the measurements and the reporting of results.

Development and evaluation of a harmonized whole body physiologically based pharmacokinetic (PBPK) model for flutamide in rats and its extrapolation to humans.

By their definition, inadvertent exposure to endocrine disrupting compounds (EDCs) intervenes with the endocrine signalling system, even at low dose. On the one hand, some EDCs are used as important pharmaceutical drugs that one would not want to dismiss. On the other hand, these pharmaceutical drugs are having off-target effects and increasingly significant exposure to the general population with unwanted health implications. Flutamide, one of the top pharmaceutical products marketed all over the world for the treatment of prostate cancer, is also a pollutant. Its therapeutic action mainly depends on targeting the androgen receptors and inhibiting the androgen action that is essential for growth and survival of prostate tissue. Currently flutamide is of concern with respect to its categorization as an endocrine disruptor. In this work we have developed a physiologically based pharmacokinetic (PBPK) model of flutamide that could serve as a standard tool for its human risk assessment. First we built the model for rat (where many parameters have been measured). The rat PBPK model was extrapolated to human where the re-parameterization involved human-specific physiology, metabolic kinetics derived from in-vitro studies, and the partition coefficient same as the rat model. We have harmonized the model by integrating different sets of in-vitro, in-vivo and physiological data into a PBPK model. Then the model was used to simulate different exposure scenarios and the results were compared against the observed data. Both uncertainty and sensitivity analysis was done. Since this new whole-body PBPK model can predict flutamide concentrations not only in plasma but also in various organs, the model may have clinical applications in efficacy and safety assessment of flutamide. The model can also be used for reverse dosimetry in the context of interpreting the available biomonitoring data to estimate the degree to which the population is currently being exposed, and a tool for the pharmaceutical companies to validate the estimated Permitted Daily Exposure (PDE) for flutamide.

Metabolic flexibility of a prospective bioremediator: Desulfitobacterium hafniense Y51 challenged in chemostats.

Desulfitobacterium hafniense Y51 has been widely used in investigations of perchloroethylene (PCE) biodegradation, but limited information exists on its other physiological capabilities. We investigated how D. hafniense Y51 confronts the debilitating limitations of not having enough electron donor (lactate), or electron acceptor (fumarate) during cultivation in chemostats. The residual concentrations of the substrates supplied in excess were much lower than expected. Transcriptomics, proteomics and fluxomics were integrated to investigate how this phenomenon was regulated. Through diverse regulation at both transcriptional and translational levels, strain Y51 turned to fermenting the excess lactate and disproportionating the excess fumarate under fumarate- and lactate-limiting conditions respectively. Genes and proteins related to the utilization of a variety of alternative electron donors and acceptors absent from the medium were induced, apparently involving the Wood-Ljungdahl pathway. Through this metabolic flexibility, D. hafniense Y51 may be able to switch between different metabolic capabilities under limiting conditions.

Learning to read and write in evolution: from static pseudoenzymes and pseudosignalers to dynamic gear shifters.

We present a systems biology view on pseudoenzymes that acknowledges that genes are not selfish: the genome is. With network function as the selectable unit, there has been an evolutionary bonus for recombination of functions of and within proteins. Many proteins house a functionality by which they 'read' the cell's state, and one by which they 'write' and thereby change that state. Should the writer domain lose its cognate function, a 'pseudoenzyme' or 'pseudosignaler' arises. GlnK involved in Escherichia coli ammonia assimilation may well be a pseudosignaler, associating 'reading' the nitrogen state of the cell to 'writing' the ammonium uptake activity. We identify functional pseudosignalers in the cyclin-dependent kinase complexes regulating cell-cycle progression. For the mitogen-activated protein kinase pathway, we illustrate how a 'dead' pseudosignaler could produce potentially selectable functionalities. Four billion years ago, bioenergetics may have shuffled 'electron-writers', producing various networks that all served the same function of anaerobic ATP synthesis and carbon assimilation from hydrogen and carbon dioxide, but at different ATP/acetate ratios. This would have enabled organisms to deal with variable challenges of energy need and substrate supply. The same principle might enable 'gear-shifting' in real time, by dynamically generating different pseudo-redox enzymes, reshuffling their coenzymes, and rerouting network fluxes. Non-stationary pH gradients in thermal vents together with similar such shuffling mechanisms may have produced a first selectable proton-motivated pyrophosphate synthase and subsequent ATP synthase. A combination of functionalities into enzymes, signalers, and the pseudo-versions thereof may offer fitness in terms of plasticity, both in real time and in evolution.

Tracing the molecular basis of transcriptional dynamics in noisy data by using an experiment-based mathematical model.

Changes in transcription factor levels, epigenetic status, splicing kinetics and mRNA degradation can each contribute to changes in the mRNA dynamics of a gene. We present a novel method to identify which of these processes is changed in cells in response to external signals or as a result of a diseased state. The method employs a mathematical model, for which the kinetics of gene regulation, splicing, elongation and mRNA degradation were estimated from experimental data of transcriptional dynamics. The time-dependent dynamics of several species of adipose differentiation-related protein (ADRP) mRNA were measured in response to ligand activation of the transcription factor peroxisome proliferator-activated receptor δ (PPARδ). We validated the method by monitoring the mRNA dynamics upon gene activation in the presence of a splicing inhibitor. Our mathematical model correctly identifies splicing as the inhibitor target, despite the noise in the data.

The Peculiar Glycolytic Pathway in Hyperthermophylic Archaea: Understanding Its Whims by Experimentation In Silico.

Mathematical models are key to systems biology where they typically describe the topology and dynamics of biological networks, listing biochemical entities and their relationships with one another. Some (hyper)thermophilic Archaea contain an enzyme, called non-phosphorylating glyceraldehyde-3-phosphate dehydrogenase (GAPN), which catalyzes the direct oxidation of glyceraldehyde-3-phosphate to 3-phosphoglycerate omitting adenosine 5'-triphosphate (ATP) formation by substrate-level-phosphorylation via phosphoglycerate kinase. In this study we formulate three hypotheses that could explain functionally why GAPN exists in these Archaea, and then construct and use mathematical models to test these three hypotheses. We used kinetic parameters of enzymes of Sulfolobus solfataricus (S. solfataricus) which is a thermo-acidophilic archaeon that grows optimally between 60 and 90 °C and between pH 2 and 4. For comparison, we used a model of Saccharomyces cerevisiae (S. cerevisiae), an organism that can live at moderate temperatures. We find that both the first hypothesis, i.e., that the glyceraldehyde-3-phosphate dehydrogenase (GAPDH) plus phosphoglycerate kinase (PGK) route (the alternative to GAPN) is thermodynamically too much uphill and the third hypothesis, i.e., that GAPDH plus PGK are required to carry the flux in the gluconeogenic direction, are correct. The second hypothesis, i.e., that the GAPDH plus PGK route delivers less than the 1 ATP per pyruvate that is delivered by the GAPN route, is only correct when GAPDH reaction has a high rate and 1,3-bis-phosphoglycerate (BPG) spontaneously degrades to 3PG at a high rate.

Multiplex Eukaryotic Transcription (In)activation: Timing, Bursting and Cycling of a Ratchet Clock Mechanism.

Activation of eukaryotic transcription is an intricate process that relies on a multitude of regulatory proteins forming complexes on chromatin. Chromatin modifications appear to play a guiding role in protein-complex assembly on chromatin. Together, these processes give rise to stochastic, often bursting, transcriptional activity. Here we present a model of eukaryotic transcription that aims to integrate those mechanisms. We use stochastic and ordinary-differential-equation modeling frameworks to examine various possible mechanisms of gene regulation by multiple transcription factors. We find that the assembly of large transcription factor complexes on chromatin via equilibrium-binding mechanisms is highly inefficient and insensitive to concentration changes of single regulatory proteins. An alternative model that lacks these limitations is a cyclic ratchet mechanism. In this mechanism, small protein complexes assemble sequentially on the promoter. Chromatin modifications mark the completion of a protein complex assembly, and sensitize the local chromatin for the assembly of the next protein complex. In this manner, a strict order of protein complex assemblies is attained. Even though the individual assembly steps are highly stochastic in duration, a sequence of them gives rise to a remarkable precision of the transcription cycle duration. This mechanism explains how transcription activation cycles, lasting for tens of minutes, derive from regulatory proteins residing on chromatin for only tens of seconds. Transcriptional bursts are an inherent feature of such transcription activation cycles. Bursting transcription can cause individual cells to remain in synchrony transiently, offering an explanation of transcriptional cycling as observed in cell populations, both on promoter chromatin status and mRNA levels.

Molecular assessment of bacterial vaginosis by Lactobacillus abundance and species diversity.

BACKGROUND: To date, women are most often diagnosed with bacterial vaginosis (BV) using microscopy based Nugent scoring or Amsel criteria. However, the accuracy is less than optimal. The aim of the present study was to confirm the identity of known BV-associated composition profiles and evaluate indicators for BV using three molecular methods. METHODS: Evaluation of indicators for BV was carried out by 16S rRNA amplicon sequencing of the V5-V7 region, a tailor-made 16S rRNA oligonucleotide-based microarray, and a PCR-based profiling technique termed IS-profiling, which is based on fragment variability of the 16S-23S rRNA intergenic spacer region. An inventory of vaginal bacterial species was obtained from 40 females attending a Dutch sexually transmitted infection outpatient clinic, of which 20 diagnosed with BV (Nugent score 7-10), and 20 BV negative (Nugent score 0-3). RESULTS: Analysis of the bacterial communities by 16S rRNA amplicon sequencing revealed two clusters in the BV negative women, dominated by either Lactobacillus iners or Lactobacillus crispatus and three distinct clusters in the BV positive women. In the former, there was a virtually complete, negative correlation between L. crispatus and L. iners. BV positive subjects showed cluster profiles that were relatively high in bacterial species diversity and dominated by anaerobic species, including Gardnerella vaginalis, and those belonging to the Families of Lachnospiraceae and Leptotrichiaceae. Accordingly, the Gini-Simpson index of species diversity, and the relative abundance Lactobacillus species appeared consistent indicators for BV. Under the conditions used, only the 16S rRNA amplicon sequencing method was suitable to assess species diversity, while all three molecular composition profiling methods were able to indicate Lactobacillus abundance in the vaginal microbiota. CONCLUSION: An affordable and simple molecular test showing a depletion of the genus Lactobacillus in combination with an increased species diversity of vaginal microbiota could serve as an alternative and practical diagnostic method for the assessment of BV.

Quantitative analysis of drug effects at the whole-body level: a case study for glucose metabolism in malaria patients.

We propose a hierarchical modelling approach to construct models for disease states at the whole-body level. Such models can simulate effects of drug-induced inhibition of reaction steps on the whole-body physiology. We illustrate the approach for glucose metabolism in malaria patients, by merging two detailed kinetic models for glucose metabolism in the parasite Plasmodium falciparum and the human red blood cell with a coarse-grained model for whole-body glucose metabolism. In addition we use a genome-scale metabolic model for the parasite to predict amino acid production profiles by the malaria parasite that can be used as a complex biomarker.

Systems Pharmacology: An opinion on how to turn the impossible into grand challenges.

A pharmacology that hits single disease-causing molecules with a single drug passively distributing to the target tissue, was almost ready. Such a pharmacology is not (going to be) effective however: a great many diseases are systems biology diseases; complex networks of some hundred thousand types of molecule, determine the functions that constitute human health, through nonlinear interactions. Malfunctions are caused by a variety of molecular failures at the same time; rarely the same variety in different individuals; in complex constellations of OR and AND logics. Few molecules cause disease single-handedly and few drugs will cure the disease all by themselves when dosed for a limited amount of time. We here discuss the implications that this discovery of the network nature of disease should have for pharmacology. We suggest ways in which pharmacokinetics, pharmacodynamics, but also systems biology and genomics may have to change so as better to deal with systems-biology diseases.

Regulation of the activity of lactate dehydrogenases from four lactic acid bacteria.

Despite high similarity in sequence and catalytic properties, the l-lactate dehydrogenases (LDHs) in lactic acid bacteria (LAB) display differences in their regulation that may arise from their adaptation to different habitats. We combined experimental and computational approaches to investigate the effects of fructose 1,6-bisphosphate (FBP), phosphate (Pi), and ionic strength (NaCl concentration) on six LDHs from four LABs studied at pH 6 and pH 7. We found that 1) the extent of activation by FBP (Kact) differs. Lactobacillus plantarum LDH is not regulated by FBP, but the other LDHs are activated with increasing sensitivity in the following order: Enterococcus faecalis LDH2 ≤ Lactococcus lactis LDH2 < E. faecalis LDH1 < L. lactis LDH1 ≤ Streptococcus pyogenes LDH. This trend reflects the electrostatic properties in the allosteric binding site of the LDH enzymes. 2) For L. plantarum, S. pyogenes, and E. faecalis, the effects of Pi are distinguishable from the effect of changing ionic strength by adding NaCl. 3) Addition of Pi inhibits E. faecalis LDH2, whereas in the absence of FBP, Pi is an activator of S. pyogenes LDH, E. faecalis LDH1, and L. lactis LDH1 and LDH2 at pH 6. These effects can be interpreted by considering the computed binding affinities of Pi to the catalytic and allosteric binding sites of the enzymes modeled in protonation states corresponding to pH 6 and pH 7. Overall, the results show a subtle interplay among the effects of Pi, FBP, and pH that results in different regulatory effects on the LDHs of different LABs.

'Domino' systems biology and the 'A' of ATP.

We develop a strategic 'domino' approach that starts with one key feature of cell function and the main process providing for it, and then adds additional processes and components only as necessary to explain provoked experimental observations. The approach is here applied to the energy metabolism of yeast in a glucose limited chemostat, subjected to a sudden increase in glucose. The puzzles addressed include (i) the lack of increase in adenosine triphosphate (ATP) upon glucose addition, (ii) the lack of increase in adenosine diphosphate (ADP) when ATP is hydrolyzed, and (iii) the rapid disappearance of the 'A' (adenine) moiety of ATP. Neither the incorporation of nucleotides into new biomass, nor steady de novo synthesis of adenosine monophosphate (AMP) explains. Cycling of the 'A' moiety accelerates when the cell's energy state is endangered, another essential domino among the seven required for understanding of the experimental observations. This new domino analysis shows how strategic experimental design and observations in tandem with theory and modeling may identify and resolve important paradoxes. It also highlights the hitherto unexpected role of the 'A' component of ATP.

Glutathione metabolism modeling: a mechanism for liver drug-robustness and a new biomarker strategy.

BACKGROUND: Glutathione metabolism can determine an individual's ability to detoxify drugs. To increase understanding of the dynamics of cellular glutathione homeostasis, we have developed an experiment-based mathematical model of the kinetics of the glutathione network. This model was used to simulate perturbations observed when human liver derived THLE cells, transfected with human cytochrome P452E1 (THLE-2E1 cells), were exposed to paracetamol (acetaminophen). METHODS: Human liver derived cells containing extra human cytochrome P4502E1 were treated with paracetamol at various levels of methionine and in the presence and absence of an inhibitor of glutamyl-cysteine synthetase (GCS). GCS activity was also measured in extracts. Intracellular and extracellular concentrations of substances involved in glutathione metabolism were measured as was damage to mitochondria and proteins. A bottom up mathematical model was made of the metabolic pathways around and including glutathione. RESULTS: Our initial model described some, but not all the metabolite-concentration and flux data obtained when THLE-2E1 cells were exposed to paracetamol at concentrations high enough to affect glutathione metabolism. We hypothesized that the lack of correspondence could be due to upregulation of expression of glutamyl cysteine synthetase, one of the enzymes controlling glutathione synthesis, and confirmed this experimentally. A modified model which incorporated this adaptive response adequately described the observed changes in the glutathione pathway. Use of the adaptive model to analyze the functioning of the glutathione network revealed that a threshold input concentration of methionine may be required for effective detoxification of reactive metabolites by glutathione conjugation. The analysis also provided evidence that 5-oxoproline and ophthalmic acid are more useful biomarkers of glutathione status when analyzed together than when analyzed in isolation, especially in a new, model-assisted integrated biomarker strategy. CONCLUSION: A robust mathematical model of the dynamics of cellular changes in glutathione homeostasis in cells has been developed and tested in vitro. GENERAL SIGNIFICANCE: Mathematical models of the glutathione pathway that help examine mechanisms of cellular protection against xenobiotic toxicity and the monitoring thereof, can now be made.

Arsenic Contamination of Groundwater Is Determined by Complex Interactions between Various Chemical and Biological Processes.

At a great many locations worldwide, the safety of drinking water is not assured due to pollution with arsenic. Arsenic toxicity is a matter of both systems chemistry and systems biology: it is determined by complex and intertwined networks of chemical reactions in the inanimate environment, in microbes in that environment, and in the human body. We here review what is known about these networks and their interconnections. We then discuss how consideration of the systems aspects of arsenic levels in groundwater may open up new avenues towards the realization of safer drinking water. Along such avenues, both geochemical and microbiological conditions can optimize groundwater microbial ecology vis-à-vis reduced arsenic toxicity.

Fusion-fission-mitophagy cycling and metabolic reprogramming coordinate nerve growth factor (NGF)-dependent neuronal differentiation.

Neuronal differentiation is regulated by nerve growth factor (NGF) and other neurotrophins. We explored the impact of NGF on mitochondrial dynamics and metabolism through time-lapse imaging, metabolomics profiling, and computer modeling studies. We show that NGF may direct differentiation by stimulating fission, thereby causing selective mitochondrial network fragmentation and mitophagy, ultimately leading to increased mitochondrial quality and respiration. Then, we reconstructed the dynamic fusion-fission-mitophagy cycling of mitochondria in a computer model, integrating these processes into a single network mechanism. Both the computational model and the simulations are able to reproduce the proposed mechanism in terms of mitochondrial dynamics, levels of reactive oxygen species (ROS), mitophagy, and mitochondrial quality, thus providing a computational tool for the interpretation of the experimental data and for future studies aiming to detail further the action of NGF on mitochondrial processes. We also show that changes in these mitochondrial processes are intertwined with a metabolic function of NGF in differentiation: NGF directs a profound metabolic rearrangement involving glycolysis, TCA cycle, and the pentose phosphate pathway, altering the redox balance. This metabolic rewiring may ensure: (a) supply of both energy and building blocks for the anabolic processes needed for morphological reorganization, as well as (b) redox homeostasis.

Testing biochemistry revisited: how in vivo metabolism can be understood from in vitro enzyme kinetics.

A decade ago, a team of biochemists including two of us, modeled yeast glycolysis and showed that one of the most studied biochemical pathways could not be quite understood in terms of the kinetic properties of the constituent enzymes as measured in cell extract. Moreover, when the same model was later applied to different experimental steady-state conditions, it often exhibited unrestrained metabolite accumulation.Here we resolve this issue by showing that the results of such ab initio modeling are improved substantially by (i) including appropriate allosteric regulation and (ii) measuring the enzyme kinetic parameters under conditions that resemble the intracellular environment. The following modifications proved crucial: (i) implementation of allosteric regulation of hexokinase and pyruvate kinase, (ii) implementation of V(max) values measured under conditions that resembled the yeast cytosol, and (iii) redetermination of the kinetic parameters of glyceraldehyde-3-phosphate dehydrogenase under physiological conditions.Model predictions and experiments were compared under five different conditions of yeast growth and starvation. When either the original model was used (which lacked important allosteric regulation), or the enzyme parameters were measured under conditions that were, as usual, optimal for high enzyme activity, fructose 1,6-bisphosphate and some other glycolytic intermediates tended to accumulate to unrealistically high concentrations. Combining all adjustments yielded an accurate correspondence between model and experiments for all five steady-state and dynamic conditions. This enhances our understanding of in vivo metabolism in terms of in vitro biochemistry.

On paradoxes between optimal growth, metabolic control analysis, and flux balance analysis.

In Microbiology it is often assumed that growth rate is maximal. This may be taken to suggest that the dependence of the growth rate on every enzyme activity is at the top of an inverse-parabolic function, i.e. that all flux control coefficients should equal zero. This might seem to imply that the sum of these flux control coefficients equals zero. According to the summation law of Metabolic Control Analysis (MCA) the sum of flux control coefficients should equal 1 however. And in Flux Balance Analysis (FBA) catabolism is often limited by a hard bound, causing catabolism to fully control the fluxes, again in apparent contrast with a flux control coefficient of zero. Here we resolve these paradoxes (apparent contradictions) in an analysis that uses the 'Edinburgh pathway', the 'Amsterdam pathway', as well as a generic metabolic network providing the building blocks or Gibbs energy for microbial growth. We review and show that (i) optimization depends on so-called enzyme control coefficients rather than the 'catalytic control coefficients' of MCA's summation law, (ii) when optimization occurs at fixed total protein, the former differ from the latter to the extent that they may all become equal to zero in the optimum state, (iii) in more realistic scenarios of optimization where catalytically inert biomass is compensating or maintenance metabolism is taken into consideration, the optimum enzyme concentrations should not be expected to equal those that maximize the specific growth rate, (iv) optimization may be in terms of yield rather than specific growth rate, which resolves the paradox because the sum of catalytic control coefficients on yield equals 0, (v) FBA effectively maximizes growth yield, and for yield the summation law states 0 rather than 1, thereby removing the paradox, (vi) furthermore, FBA then comes more often to a 'hard optimum' defined by a maximum catabolic flux and a catabolic-enzyme control coefficient of 1. The trade-off between maintenance metabolism and growth is highlighted as worthy of further analysis.